Iterating Symmetric Extensions

I don’t usually like to write about new papers. I mean, it’s a paper, you can read it, you can email me and ask about it if you’d like. It’s there. And indeed, for my previous papers, I didn’t even mention them being posted on arXiv/submitted/accepted/published. This paper is a bit different; but don’t worry, this is not your typical “new paper” post.

If you don’t follow arXiv very closely, I have posted a paper titled “Iterating Symmetric Extensions“. This is going to be the first part of my dissertation. The paper is concerned with developing a general framework for iterating symmetric extensions, which oddly enough, is something that we didn’t really know how to do until now. There is a refinement of the general framework to something I call “productive iterations” which impose some additional requirements, but allow greater freedom in the choice of filters used to interpret the names. There is an example of a class-length iteration, which effectively takes everything that was done in the paper and uses it to produce a class-length iteration—and thus a class length sequence of models—where slowly, but surely, Kinna-Wagner Principles fail more and more. This means that we are forcing “diagonally” away from the ordinals. So the models produced there will not be defined by their set of ordinals, and sets of sets of ordinals, and so on.

One seemingly unrelated theorem extends a theorem of Grigorieff, and shows that if you take an iteration of symmetric extensions, as defined in the paper, then the full generic extension is one homogeneous forcing away. This is interesting, as it has applications for ground model definability for models obtained via symmetric extensions and iterations thereof.

But again, all that is in the paper. We’re not here to discuss these results. We’re not here to read some funny comic with a T-Rex and a befuddled audience either. We’re here to talk about how the work came into fruition. Well, parts of that process. Because I feel that often we don’t talk about these things. We present the world with a completed work, or some partial work, and we move on. We don’t stop to dwell on the hardship we’ve endured. We assume, and probably correctly, that most people have endured similar difficulties one time or another. So there is no need to explain, or expose any of the background details. Well. Screw that. This is my blog, and I can write about it if I want to. And I do.

So, the idea of iterating symmetric extensions came to me when I was finishing my masters, I was thinking about a way to extend symmetric extensions, because it seemed to me that we ran this tool pretty much into the ground, and I was looking for a tool that will enable us to dig deeper into the world of non-AC models. It was a good timing, too. Menachem [Magidor] had told me about this interesting model that they constructed in Bristol at some workshop, and it seemed like a good test subject (dubbed “The Bristol Model” from that point onward). When I settled into this idea, and Menachem explained to me the vague details of the construction, it immediately seemed to me as an iteration of symmetric extensions. So I set on to develop a method that will enable me to formalize and reconstruct this model (I did that, and while I have a set of notes with a written account, I will soon start transforming those into a proper paper, so I hope that by the end of July I will have something to show for).

The first idea came to me when I was in Vienna in September of 2013. I was sure it’s going to work easy peasy, and so I left it to focus on other issues of the hour. When I came back to this a few months later, Menachem and I talked about it and identified a few possible weak spots. Somehow we managed to convince ourselves that this is not a real issue, and I started working the details. Headstrong and cocksure, I was certain there just a few small technical details which will be solved in a couple of days worth of work. But math had other plans, and I spent about a year and a half before things worked out.

Specifically because I kept running into small problems. Whenever I wrote about some statement that it’s “clear” or “obvious”, there were troubles with that later. Whenever I was sure that something has to be true, it turned out to be false. And I had to rewrite my notes many times over. Usually more or less from scratch. Luckily for me, Martin Goldstern was visiting Jerusalem for a few months during the spring semester of 2015, and he was kind enough to hear my ideas and point a lot of these problems. “Oh, just make sure that such and such is true” he would say, and the next day I’d find him and say something along the lines “Yeah, it turned out that it’s false, so I had to do this and that to circumvent the problem, but now it simplified these proofs”. And the process repeated itself. This long process is one of the great sources for this blog post of mine, and this post and also that post.

Closing in on the summer, Yair [Hayut] was listening to whatever variant I had at the time, and at some point he disagreed with one of the things I had to say. “Surely you can’t disagree with this theorem, it only relies on the lemma that I showed you as the first lemma, and you’ve agreed to that”. He pondered a little bit, and said “No, I actually disagree with the lemma”. We paused, we thought about it, and we came up with one or two counterexamples to that lemma. It was exactly the issue Menachem and I identified, and suddenly all the problems that were plaguing me because obvious consequences of that very problem.

I had worked very hard over the course of the next two months, and I managed to salvage the idea from oblivion. It was a good thing, too, because shortly after I’d visit the Newton Institute, and I had the chance to present this over the course of 8 hours to whomever was interested. And a few people were. But the definition was just terrible. I was happy it’s working, though, so I left it aside to cool down for a bit, while I worked on other projects of my thesis.

And now, I sat down to write this paper. And as I was writing it, I realized how to simplify some of the horrible details, which is great. This caused some of the proofs to be clearer, better and more of what you’d expect of these proofs. And that’s all I ever wanted, really. It took me two years, but it feels good to be over with, I hope. Now we wait for the referee report… and a year from now, when I’ve forgotten all about this, I’ll probably grunt, groan, and revise the damn thing, when the report will show up. Or maybe sooner.

Okay, maybe this sounds like I’m treating this as a rare process. And to some extent, it is. This is my first big piece of research. You can only have one first of those. Yes, mathematical research is a process. A long and slow process. I’m not here to complain about this, or argue otherwise. I’m here to point out the obvious, and complain that I never heard people talking about these sort of slow processes. Only about the “So he hopped on a plane, came over here, and we banged this thing together in a couple of weeks time”, which is really awesome and sort of exciting. But someone has to stand up and say “No, this was a slow and torturous process that drained the life out of me for the better part of two years”.